Summary: How to Compress Interactive Communication
Boaz Barak # Mark Braverman + Xi Chen # Anup Rao §
November 10, 2009
Abstract
We describe new ways to simulate 2­party communication protocols to get protocols with
potentially smaller communication. We show that every communication protocol that commu­
nicates C bits and reveals I bits of information to the participating parties can be simulated by
a new protocol involving at most ”
O( # CI) bits of communication. In the case that the parties
have inputs that are independent of each other, we get much better results, showing how to
carry out the simulation with ”
O(I) bits of communication.
These results lead to a direct sum theorem for randomized communication complexity. Ig­
noring polylogarithmic factors, we show that for worst case computation, computing n copies
of a function requires # n times the communication required for computing on copy of the func­
tion. For average case complexity, given any distribution µ on inputs, computing n copies of the
function on n independent inputs sampled according to µ requires # n times the communication
for computing one copy. If µ is a product distribution, computing n copies on n independent
inputs sampled according to µ requires n times the communication required for computing the
function. We also study the complexity of computing the sum (or parity) of n evaluations of f ,